Consistency of Genera lized Maximum Spacing Estimates

General methods for the estimation of distributions can be derived from approx­ imations of certain information measures. For example, both the maximum like­ lihood (ML) method and the maximum spacing (MSP) method can be obtained from approximations of t he Kuliback-Leibler information. The ideas behind the MSP method, whereby an estimation method for continuous univariate distri­ butions is obtained from an approximation based on spacings of an information measure, were used by Ranneby and Ekström (1997) (using simple spacings) and Ekström (1997) (using high order spacings) to obtain a class of e stimation methods, called generalized maximum spacing (GMSP) methods. In the present paper, GMSP methods will be shown to give consistent estimates under general conditions, comparable to those of Bahadur (1971) for the ML method, and those of S hao and Hahn (1996) for the MSP method. In particular, it will be shown that GMSP methods give Ll consistent estimates in any family of distributions with unimodal densities, without any further conditions on the distributions.